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Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

100%

McKee J., Smyth C.

Open Mathematics

2013

tom 11

nr 5

882-899

We give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also corresponds (in four different ways) to a single Pisot polynomial from a certain restricted class, the cyclogenic Pisot polynomials. We investigate properties of this class of Pisot polynomials.

Comments on the height reducing property

64%

Akiyama S., Zaimi T.

Open Mathematics

2013

tom 11

nr 9

1616-1627

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.

Ograniczanie wyników

2 Open Mathematics

1 Akiyama S.

1 McKee J.

1 Smyth C.

1 Zaimi T.

2 2013

Wyniki wyszukiwania

Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

Comments on the height reducing property